#### DOCUMENTA MATHEMATICA, Vol. 20 (2015), 507-529

Elena Angelini

Logarithmic Bundles of Multi-Degree Arrangements in P^n

Let $D = {D_{1}, \ldots, D_{\ell}}$ be a multi-degree arrangement with normal crossings on the complex projective space $\{P}^n$, with degrees $d_{1}, \ldots, d_{\ell}$; let $\Omega_{\{P}^n}^1(\log D)$ be the logarithmic bundle attached to it. First we prove a Torelli type theorem when $D$ has a sufficiently large number of components by recovering them as unstable smooth irreducible degree-$d_{i}$ hypersurfaces of $\Omega_{\{P}^n}^1(\log D)$. Then, when $n = 2$, by describing the moduli spaces containing $\Omega_{\{P}^2}^1(\log D)$, we show that arrangements of a line and a conic, or of two lines and a conic, are not Torelli. Moreover we prove that the logarithmic bundle of three lines and a conic is related with the one of a cubic. Finally we analyze the conic-case.

2010 Mathematics Subject Classification: 14J60, 14F05, 14C34, 14C20, 14N05

Keywords and Phrases: Multi-degree arrangement, Hyperplane arrangement, Logarithmic bundle, Torelli theorem

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